1. Introduction

Suspensions of spherical colloidal particles have proven a valuable model system for understanding many complex phenomena. Perhaps most notably, the model has provided insight into dynamical processes within different phases of matter [1–8], as well as various mechanisms involved during phase transitions [9–14]. A combination of digital video microscopy and computerized particle tracking algorithms [15, 16] has allowed for the direct visualization of such colloidal systems and measurement of static and dynamical properties under many experimental conditions [17–21]. Given the spherical symmetry of the particles, most previous studies have understandably focused only on understanding translational dynamics, though there have been a few studies on rotational dynamics in dilute systems of spheres [22, 23], and on the translational and rotational dynamics of anisotropic colloidal particles [24–27].

Within the past decade, researchers have developed a variety of techniques for synthesizing clusters of colloidal particles with a wide range of reproducible morphologies [28, 29], several of which are shown in Fig. 1. Colloidal clusters have the potential to extend the colloidal model past one of simple spherical particles and into a realm where the collective dynamics of particles with complex shapes, more representative of molecules, can be studied [30–32].

 

Fig. 1 Top: Volumetric images of colloidal clusters with n = 4, 5, and 6 from 3D confocal micrographs. Images have been filtered and enhanced to allow easy visualization of the 3-dimensional structures. Individual particles are approximately 2 μm in diameter. Bottom: Ball-and-stick models of the clusters above. The cases n = 4 and n = 5 are accurate representations of the simulated tetrahedra and pentahedra discussed in the text.

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Conventional particle tracking methods [15, 16] are designed to follow the translational motions of individual particles, and so are immediately applicable to colloidal clusters, provided the particles can be reliably distinguished. However, a description of dynamics within systems composed of or including clusters is incomplete without knowledge of how rotational degrees of freedom are explored. Such an understanding could provide further insight into fundamental behaviors of systems with orientational order, such as liquid crystals, or systems subjected to external fields and anisotropic flows [27]. To our knowledge, there has been only one study that combined video microscopy with particle tracking to measure the rotational motion of colloidal clusters [33], using different methods than those described in this work. That study focused on dilute systems of planar clusters undergoing two-dimensional diffusion near a boundary, and revealed a decoupling of translational and rotational diffusion due to hydrodynamic effects.

We present here a simple and generalized method to track and calculate the two- or three-dimensional rotational motions of clusters of colloidal particles. Our method uses existing particle tracking routines and rigid body transformations to measure the changes in orientation of a cluster over time. To demonstrate the effectiveness and accuracy of the method, we simulate the motion of colloidal clusters under conditions that are relevant to microscopy experiments. We then use our method to measure the rotational diffusion coefficient of a real colloidal cluster.

2. Calculating rotations

Our procedure for calculating rotational displacements of clusters is based on a method by Challis for determining rigid body transformations between reference frames [34]. Challis’ procedure was originally intended for comparing osteometric measurements in biomechanical analyses. As we will show, the method can be intuitively adapted to measure rotational dynamics. First, we give a brief reprise of Challis’ method, and then describe how it is used to study the systems mentioned here.

2.1. Challis’ procedure for coordinate transformations

Given a set of points which have coordinates {x _i} measured in one reference frame and coordinates {y _i} measured in a second frame, there exists a transformation

We are therefore left with

For a set of n points, R can be calculated using a least squares approach. This method minimizes the quantity

At this point, a singular value decomposition (SVD) is performed on C such that

2.2. Application to colloidal clusters

Particle tracking [15, 16] yields vector coordinates x′_i for every particle i over a distinct set of times. Hence, the first step in tracking rotational motion of a cluster is to track the translational motion of each particle within the cluster. For each cluster, we first determine the center of mass x _CM at a given time and subtract this quantity from the coordinates of particles belonging to the cluster, thereby removing any translational motion. We are left with new coordinates x _i in the center of mass frame,

Therefore, given the coordinates from a particle tracking experiment, we may use Challis’ procedure, starting from Eq. (8), to calculate a cross dispersion matrix C and subsequently perform a singular value decomposition to calculate a rotation matrix for each pair of successive times [t,t + Δt]. With the complete set of rotation matrices, {R ^k}, we may reconstruct the entire trajectory of a particle about the cluster center of mass by computing the product of successive rotations. Given ${\mathbf{\text{x}}}_i^0$, the position of a particle at some later time t can be calculated as

The advantage of calculating {R^k} is that it describes the collective behaviors of particles within a cluster, rather than a property of any individual particle. For example, knowledge of {R^k} for a cluster allows for immediate calculation of the motions of any particular particle about the center of mass, or the motion of the cluster about any arbitrary axis of rotation. Diffusive anisotropic clusters with large aspect ratios rotate more freely about a long axis than about a short axis. Given {R^k}, however, one needs only the initial orientation of these axes to compute and compare the motions around them. An additional advantage of this procedure is that it is computationally fast, and the necessary linear algebra routines are standardly included in most mathematical software suites.

3. Tests of the prescribed method

For simplicity and to better reproduce the data collection process in typical microscopy experiments, we adopt a length scale of microns and a time scale measured in timesteps (ts), which is equivalent in microscopy to video frames or image stacks.

In conventional particle tracking experiments, it is important to minimize and understand the uncertainty, i.e. the noise, inherent in locating a particle. Typical microscopy experiments combine high magnification optics with CCD cameras to record raw digital images. In the absence of other sources of noise, the uncertainty in particle position, i.e. the minimum noise level, depends on the optical resolution [pixels/distance] of the instrumentation and the size [pixels] of the object being tracked. Standard image processing and particle tracking techniques can locate the centers of particles to within ≈ 1/N of a pixel, where N is the width of the object in pixels. Camera resolution varies between experimental set-ups but is typically in the range of 0.2 μm/pixel. The minimum uncertainty in particle position is the product of these factors. For example, observing a 10 pixel wide object with a resolution of 0.2 μm/pixel leads to a lower limit of ≈ 20 nm uncertainty in particle position. Other sources of noise, such as stray light entering the microscope, noise within the CCD camera itself, etc., slightly increase the uncertainty in particle position and further limit particle tracking resolution.

To test our method, we simulate the rotational Brownian motion of tetra- and pentahedral clusters with different rotational diffusion coefficients, D_R, and different levels of noise, σ_x. We first generate noise-free cluster trajectories. For tetrehedra, we place particles at initial coordinates

At each time step, we select three random angles, α, β, γ, from a Gaussian distribution with a standard deviation of $\sqrt{2D_R}$. This distribution ensures that the simulated dynamics will be in agreement with the Stokes-Einstein-Debye relation discussed later [Eq. (20)]. Each particle in a tetrahedron is rotated by an angle γ about the z-axis, then by an angle β about the y-axis, and finally by angle α about the x-axis to produce the tetrahedron at the subsequent time. The rotation matrices used are

This method of generating rotational Brownian motion is not strictly correct for the given pentahedral geometry as its motions cannot be generally characterized by a single D_R. However, using the publicly available package HYDRO++ [35], we calculate the difference in D_R between long and short axes to be ≈ 2%. Given this small variation, the use of a single D_R to describe a pentahedron is a more than sufficient approximation for the purpose of testing our algorithm, which after all does not require that the motions agree with the laws of diffusion.

Trajectories generated in this way are noise-free in the sense that they are absent of uncertainty in particle position to within machine precision. To mimic the type of experimental noise previously mentioned, we post-process the trajectories by adding Gaussian random numbers, with standard deviation σ_x, to the particle coordinates. The levels of noise presented here correspond to uncertainties of σ_x ∈ {10, 30, 50, 70, 100} nm in the x-, y-, and z-directions. Experimental uncertainties are typically within the range of 20-60 nm, and so the levels studied here are relevant to microscopy experiments. After the noise is added, we apply our method of measuring rotational motion in order to gauge the effect of experimental noise on results.

4. Analysis of rotational motion

In this section, we calculate the rotational motions of simulated colloidal clusters using the matrix methods described above. We focus on a method that uses the rotation matrices to determine the motion of a fictional orientation vector attached to the cluster. The rotational displacements of one or more such vectors about the center of mass can be used to compare and contrast motions about different axes, which is especially useful when the objects studied are anisotropic and motions about axes are expected to differ [36, 37].

We note that a second method exists to calculate rotational displacements using solely the rotation matrix. This method determines the rotational axis from R and can then be used to calculate the magnitude of an angular displacement. While a mathematically direct and general approach, it describes only the average cluster dynamics and can provide no insight into how motions about different axes vary. Given these limitations, we relegate a description of this method to the appendix.

To analyze the rotations of a cluster, we consider the motions of an initial orientational unit vector p̂ ⁰ fixed to the cluster. We determine its orientation at a later time t by applying the set of rotations such that

In line with [36–38], we define a vector rotational displacement

To illustrate the meaning of φ⃗(t), consider an object with constant angular velocity ω⃗ = ω ẑ. Over a time Δt, the rotational vector displacement is given by φ⃗ (Δt) = ω⃗Δt = (0, 0, ωΔt). Therefore in general, φ⃗(t) has components in each of the Cartesian axes, (φ_x, φ_y, φ_z), corresponding to cumulative rotations about those axes.

In Fig. 2(a), we show the orientational trajectories of particles within a simulated tetrahedron projected onto the surface of a unit sphere. Fig. 2(b) shows 2D projections of trajectories for two p̂ ⁰ through the rotation space described above. As shown in Fig. 2, even though the cluster is a solid body, trajectories of individual particles differ due to rotations of the cluster about random axes.

 

Fig. 2 (a) Trajectories of particles within a simulated Brownian tetrahedral cluster projected onto the surface of a unit sphere. Colors represent different particles, i.e. different choices for p̂ ⁰. (b) 2d projection of two trajectories through rotation space (φ_x, φ_y). Colors correspond to the same particles in (a). Both trajectories in (b) begin at (0, 0) and end at open circles.

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Given the definition of φ⃗(t) in Eq. (18), we may define an unbounded mean square angular displacement (MSAD), akin to a translational mean square displacement, as

In Fig. 3(a) & (b), we show the MSAD of two simulated tetrahedral clusters with diffusion constants 10⁻⁴ rad²/ts and 10⁻³ rad²/ts respectively and different levels of noise. The influence of noise is apparent in Fig. 3 as deviations from linearity at small Δt. Eventually, the MSAD recovers the true diffusive behavior because the cluster has made rotations large enough to distinguish from the noise.

 

Fig. 3 Mean square angular displacements of simulated tetrahedral clusters ( $R\hspace{0.17em}=\hspace{0.17em}\sqrt{3}\hspace{0.17em}\mu \text{m}$) for different noise levels σ_x and diffusion coefficients of (a) 10⁻⁴ rad²/ts and (b) 10⁻³ rad²/ts. Open circles are the theoretical MSAD based on Eq. (20). Deviations from linearity at small Δt demonstrate the effect of noise when resolving small rotations. Deviations at large Δt, however, are the result of low statistics at these lag times.

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With no experimental noise, a log-log plot of MSAD as a function of Δt will be a straight line with a slope of unity, as indicated by the open circles in Figs. 3(a) & (b). However, in the presence of noise in particle positions, rotations cannot be accurately resolved below a certain threshold, Φ. For example, a stationary cluster will appear to make small, but fictional, rotations as a result of this noise, and measurements of the MSAD will yield

In the case of translational diffusion, one expects that the noise in each direction will contribute an error of $2{\sigma }_x^2$ to the mean-square displacement [15]. In terms of an angular uncertainty, this contribution is diminished by a factor of R ², where R is the average distance from a particle to the cluster center of mass. Our matrix method reduces this uncertainty further by a factor of n. Thus, when only static noise is present, one expects for a diffusing cluster

To test this assertion, we add noise to simulations of stationary (D_R = 0) tetrahedral clusters and calculate the MSADs. As in Eq. (21), the MSADs are constant over time. We take Φ² to be the value of the MSAD as Δt → 0. Shown in Fig. 4(a), Eq. (23) accurately describes the static angular uncertainty for a wide range of tetra- and pentahedral cluster sizes, R, and noise levels. Fig. 4(b) shows all of the previously measured MSADs in Fig. 3 plotted with the noise subtracted. This precisely collapses the MSADs to the true values in each case.

 

Fig. 4 (a) Measured values of Φ² for non-diffusing tetrahedral (solid symbols) and pentahedral (open symbols) clusters. Colors indicate noise levels and symbols indicate cluster size. Solid lines are the prediction ${\mathrm{\Phi }}^2\hspace{0.17em}=\hspace{0.17em}6{\sigma }_x^2/n{R}^2$. (b) Same data in Fig. 3 with the appropriate Φ² for each noise level subtracted.

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Knowing how uncertainty in particle positions affects measurements of dynamics quantities, such as the MSAD, is clearly important. However, one is typically unable to define the noise level so precisely in experiments. As stated earlier, particle tracking resolution depends on various experimental factors including particle size, optics, and the type of camera used to record images. As shown in Fig. 3(a), for slowly diffusing clusters and moderate-to-high noise levels, the measurements of the MSAD may not be representative of the true dynamics until fairly large lag times.

In microscopy experiments, one can effectively increase the rotational signal-to-noise ratio between each time step by imaging less frequently. Determining the appropriate sampling rates in experiments can be done by estimating the noise level σ_x, the diffusion constant D_R and calculating an approximate Φ². Diffusive motion will begin exceeding noise when Φ² ≈ 4D_RΔt. Solving for Δt then yields a reasonable sampling lag time. It is also important to avoid under-sampling, that is, too long a lag time. Doing so will make diffusive motion between images appear erroneously slow. We find that a good rule of thumb for the upper limit on sampling time should be the time when the cluster has diffused ≈1 radian². Thus, an estimate for the upper limit is ≈ 1/4D_R. However, as in all particle tracking experiments, the time between images must be small enough that individual particles can be confidently identified. This typically means that particles must be imaged before moving a distance of one interparticle spacing [15].

5. Experimental application

Measurements of rotational motion of diffusive tetrahedral clusters have been performed using the described method [29]. Real fluorescent tetrahedral clusters are synthesized as in [29]. A cluster is composed of individual poly(methyl methacrylate) (PMMA) spheres, each with a diameter 2.45 μm as measured by static light scattering (SLS). The particles within a cluster are irreversibly bound together, but are sterically stabilized to prevent the possibility of aggregation to other clusters. Dilute suspensions of clusters are prepared in a mixture of cyclohexyl bromide (CXB) and cis-decalin (DCL) at a ratio of 85/15 (w/w) that closely matches both the density and index of refraction of the particles. Clusters are imaged in 3D over time with a Leica TCS SP5 confocal microscope. We track locations of the individual particles within a tetrahedron using standard particle tracking routines [15,16]. The uncertainties in particle position for these experiments are ≈ 30 nm in the x- and y-directions, and ≈ 40 nm in the z-direction. Given these tracking resolutions and assuming a maximally packed tetrahedron, from Eq. (23) we estimate the angular resolution in this experiment as Φ ≈ 0.028 radians (1.6°).

Once tracked, we calculate the translational MSD and the MSAD and determine the translational and rotational diffusion coefficients, D_T and D_R, respectively. In three dimensions, the translational MSD is described by the Stokes-Einstein-Sutherland equation

Hydrodynamically, tetrahedral clusters can be accurately modeled as spheres [41] given the relation

In Fig. 5, we show the MSD and MSAD of a diffusing tetradedral cluster. The MSD corresponds to motions of the center of mass of the cluster, and the MSAD is an average over the individual particle MSADs (i.e., using the initial orientations of the particles as four separate p̂ ⁰). Solid lines are fits to the data over the indicated range 3 s ≤ Δt ≤ 22 s. As can be seen, both the MSD and MSAD are approximately linear only at small lag times. This is because the data set used to make these measurements consists of only a single cluster for less than 300 timesteps (≈ 220 s). Diffusion coefficients are measured by performing linear fits to the MSD and MSAD shown in Fig. 5 over the indicated range and extracting the slopes of the lines.

 

Fig. 5 Calculated MSD (blue squares) and MSAD (red circles) of a diffusing experimental tetrahedral cluster. Solid lines are best fit lines over range of data used to determine the respective diffusion coefficients.

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Shown in Table 1 are the measured translational and rotational diffusion coefficients. Values of d _tetra are calculated from Eqs. (26) & (27) using known experimental conditions, and values for d _sphere then follow trivially using Eq. (25). Ideally, the sizes calculated from translational motions would be identical to those calculated from rotational motions. As shown, these values agree to within 4%. Such good agreement between these two measurements demonstrates our ability to track translational and rotational motions of clusters simultaneously.

Table 1. Measurements from Tracking Tetrahedral Cluster

View Table

Previous work using standard particle tracking and our matrix method reported particle diameters ≈ 14% larger than the 2.45 μm measured by SLS [29]. A reanalysis of the data with improved particle tracking, presented above, increases the accuracy of the particle diameter measured using our matrix method to within ≈ 9%. The remaining disagreement is likely due to multiple factors, including the fitting routines used in SLS and swelling of particles when in a solvent of CXB/DCL. SLS measurements were performed with suspensions in pure DCL, which is not known to swell particles.

6. Conclusion

We have presented a simple method of tracking the rotational motions of colloidal clusters. Our method implements conventional particle tracking routines to determine the locations of individual particles within a cluster and uses this information to compute rigid body transformations that describe changes in a cluster’s orientation over time. The set of matrix transformations constitutes a global description of a cluster’s motion during the course of an experiment or simulation and allows one to calculate rotational dynamics about any arbitrary axis. The least squares minimization used in this method considers the motions of each individual particle in calculating the rotation matrix, and therefore measurements of rotational displacements are less sensitive to tracking noise. Additionally, precision in determining angular displacements increases with cluster size R. When combined, the resolution of measuring angular displacements scales with 1/nR ².

Though constructed specifically for tracking colloidal clusters, we emphasize that our method is not limited to these systems, but is applicable to following the rotational motion of any body over time, providing that at least three distinct noncolinear points in the body can be reliably distinguished. Because the accuracy of tracking a rotation depends on the number of particles and the size of the body, the possibility exists that this method can be adapted to many-body systems where each particle diffuses independently of the rest while the system itself also undergoes bulk rotations. In this scenario, diffusive motions are random and can therefore be treated as noise while calculating {R ^k} for the system. The rotation matrices can be inverted and, in a manner similar to Eq. (13), the bulk rotations can be removed, leaving only uncorrelated diffusive motion. Thus, this method is also applicable to determining particle motions in rotating coordinate frames.

Appendix

Given a rotation matrix R, one can calculate at each time the axis of rotation û and angular displacement Δφ. In this notation, a rotational displacement can be described by a vector Δφ⃗ = Δφû, where û has components in each of the Cartesian axes.

A single rotation will, by definition, have no effect on the direction of û, therefore,

(28)$$\mathbf{\text{R}}\widehat{\mathbf{\text{u}}}\hspace{0.17em}=\hspace{0.17em}\widehat{\mathbf{\text{u}}}.$$

From Eq. (28), we see that the axis of rotation is an eigenvector of the matrix R with an eigenvalue of 1. For a set of rotational displacements {R ^k}, one may determine the axes of rotation by calculating the eigenvectors and eigenvalues of the rotation matrices, searching for the eigenvalues equal to 1, and taking the corresponding eigenvectors.

To determine the size of the displacement about the axis of rotation, one defines an arbitrary vector ŵ perpendicular to û. For simplicity, we choose ŵ to be perpendicular to the x-axis (denoted by î),

(29)$$\widehat{\mathbf{\text{w}}}\hspace{0.17em}=\hspace{0.17em}\frac{\widehat{\mathbf{\text{u}}}\times \widehat{\mathbf{\text{i}}}}{|\widehat{\mathbf{\text{u}}}\times \widehat{\mathbf{\text{i}}}|}.$$

and apply the rotation matrix,

(30)$$\mathbf{\text{R}}\widehat{\mathbf{\text{w}}}\hspace{0.17em}=\hspace{0.17em}\widehat{\mathbf{\text{w}}}\prime .$$

The magnitude of the displacement is the angle between ŵ and ŵ′, and can be computed using the cross product relation

(31)$$\text{sin}\hspace{0.17em}(\mathrm{\Delta }\phi )\widehat{\mathbf{\text{u}}}\hspace{0.17em}=\hspace{0.17em}\widehat{\mathbf{\text{w}}}\times \widehat{\mathbf{\text{w}}}\prime .$$

Calculating displacements relative to the axis of rotation always results in displacements greater than or equal to those measured relative to an arbitrary p̂ ⁰. For example, measuring a diffusion coefficient of a spherically symmetric body with this method will yield a value that is a factor of 3/2 of the actual diffusion coefficient in Eq. (20).

To understand the origin of this difference, consider the rotational axis û = ẑ, a perpendicular vector ŵ = x̂, and an arbitrary vector p̂ ⁰ located at Cartesian coordinates (sinθ, 0, cosθ), where θ is the spherical polar angle. If ŵ is rotated by an amount δφ_u, the angle between ŵ and ŵ′ is identically δφ_u. However, the angle δφ between p̂ and p̂′ can be shown to be

(32)$$\delta \phi \hspace{0.17em}=\hspace{0.17em}{\text{cos}}^{-1}\hspace{0.17em}({\text{sin}}^2\hspace{0.17em}\theta \hspace{0.17em}\text{cos}\hspace{0.17em}\delta {\phi }_u\hspace{0.17em}+\hspace{0.17em}{\text{cos}}^2\hspace{0.17em}\theta )$$

For small δφ_u, we can approximate δφ as

(33)$$1\hspace{0.17em}-\hspace{0.17em}\delta {\phi }^2\hspace{0.17em}\approx \hspace{0.17em}{\text{sin}}^2\hspace{0.17em}\theta \hspace{0.17em}(1-\delta {\phi }_u^2)\hspace{0.17em}+\hspace{0.17em}(1\hspace{0.17em}-\hspace{0.17em}{\text{sin}}^2\hspace{0.17em}\theta )$$

(34)$$\delta {\phi }^2\hspace{0.17em}\approx \hspace{0.17em}{\text{sin}}^2\hspace{0.17em}\theta \delta {\phi }_u^2.$$

If an average of Eq. (34) is taken over spherical coordinates, we are left with an expression similar to an MSAD,

(35)$$〈\delta {\phi }^2〉\hspace{0.17em}=\hspace{0.17em}〈{\text{sin}}^2\hspace{0.17em}\theta 〉\hspace{0.17em}〈\delta {\phi }_u^2〉\hspace{0.17em}=\hspace{0.17em}\frac{2}{3}〈\delta {\phi }_u^2〉,$$

The factor of 2/3 arises for the same reasons in Perrin’s original derivation of rotational diffusion [42]. Fig. 6 shows the MSADs measured in this way for two simulated pentahedral clusters, using a single diffusion constant for each. As stated, one can correct for the overestimation of motion by simply multiplying the MSAD by 2/3. The average MSAD for any body can be corrected in the same manner, however we stress that this method returns only the average dynamics. While more direct than using an orientation vector, characterizing anisotropic bodies in this way will convolute motions about separate axes, and so will require some care when interpreting results.

 

Fig. 6 The MSADs of simulated pentahedra ( $R\hspace{0.17em}=\hspace{0.17em}\sqrt{3}\mu m$) with diffusion coefficients D_R = 10⁻⁴ rad²/ts and D_R = 10⁻² rad²/ts and σ_x = 50 nm. Dashed lines show the uncorrected MSAD described below and solid lines are corrected data, obtained by multiplying the dashed lines by 2/3. Open circles are the theoretical MSAD from Eq. (20). We note that we have approximated diffusive motion by a single D_R for each pentahedron, as described in section 3.

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Acknowledgments

G.L.H., K.V.E., and E.R.W. acknowledge financial support from NSF Grant CHE-0910707, and M.T.E. acknowledges support from NSF Grant DMR-0706453. We thank D.J. Pine for helpful discussions.

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